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Spherical Trigonometry Fundamentals

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Spherical Law of Sines

The spherical law of sines states that for any spherical triangle with sides a, b, c and corresponding opposite angles A, B, C:

\frac{\sin(a)}{\sin(A)} = \frac{\sin(b)}{\sin(B)} = \frac{\sin(c)}{\sin(C)}

Angle-Side-Angle (ASA) Problem

Given two angles and the side between them in a spherical triangle, determine the other two sides and the third angle. Applying the spherical law of sines and cosines can solve for the unknowns.

Side-Side-Side (SSS) Problem

Given the lengths of all three sides of a spherical triangle, determine the angles. Using the spherical law of cosines:

\cos(a) = \cos(b) \cos(c) + \sin(b) \sin(c) \cos(A)

solve for each angle A, B, and C.

Polar Triangle

A polar triangle is associated with a given spherical triangle, and its vertices are the poles of the sides of the original triangle. The sides of the polar triangle are the supplements of the angles of the original triangle, and vice versa.

Spherical Law of Cosines

For a spherical triangle with sides a, b, c (measured in angular units) opposite to angles A, B, C, respectively, the spherical law of cosines states:

\cos(a) = \cos(b)\cos(c) + \sin(b)\sin(c)\cos(A)

Similar formulas hold for cos(b) and cos(c).

Napier's Rules for Right Spherical Triangles

Napier's rules provide a mnemonic for solving right spherical triangles. For a right triangle with hypotenuse c opposite the right angle, circumjacent parts a and b, and adjacent angles A and B:

\tan(a) = \cot(B) \cdot \cos(c), \quad \tan(b) = \cot(A) \cdot \cos(c)

and so forth, for all circular parts of the triangle.

Spherical Excess

The spherical excess is the amount by which the sum of the angles of a spherical triangle exceeds 180°. It is denoted by E and can be used to find the area of the triangle: $A = r^2E$ , where $r$ is the radius of the sphere.

Define a spherical triangle.

A spherical triangle is a figure on the surface of a sphere, formed by three great circle arcs, which are the intersections of the sphere with three planes that pass through the sphere's center.

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